A graph \(H\)  is a minor of \(G\)  if \(H\)  can be obtained from \(G\)  by repeatedly doing one of three operations: deleting an edge, contracting an edge or deleting a vertex. We keep our graphs simple and remove any parallel edges that may form during contractions. It can be verified [1]} that a graph \(H\)  with \(k\)  vertices is a minor of \(G\) 
if and only if there are \(k\)  vertex disjoint connected subgraphs of \(G,C_1 \ldots C_k\)  such that for every edge \((v_i,v_j)\)  of \(H\) , there is an edge connecting a vertex in \(C_i\)  to a vertex of \(C_j\) .
We refer to the map mapping every vertex of \(H,v_j\)  to \(C_j\)  as an embedding of \(H\)  in \(G\) ; the maximum vertex cardinality of \(C_i,1\le i \le k\)  is called the width of the embedding. We shall be relying on the well-known fact [1]}, [3]} that if \(H\)  is a minor of \(G\)  than the treewidth of \(G\)  is lower bounded by the treewidth of \(H\) .
